Math 676: Moduli Spaces

WHEN: MWF 12 – 12:50 pm

WHERE: ENGRG E 103// in the oval with good weather

WHAT: Some good fun with moduli spaces

Moduli spaces are ubiquitous and
overwhelmingly important objects in mathematics. Simply because when we have
collections of mathematical objects that are similar, we just can’t
content of looking at them one-by-one. We want to somehow see the complete
collection at once with some sort of structure that tells us when the objects
are more or less similar to each other. A moduli space achieves such a goal.

In this course, I would like to make a wide
exploration of moduli spaces, with the goal of addressing these four main
philosophical questions:

1.What are moduli
spaces and how should we think of them?

2.How do you
construct a moduli space?

3.How do you
compactify a moduli space?

4.How do you
work with moduli spaces?

The course will be driven by concrete
examples, and weave in and out of these four points in such a way that
hopefully, by the end of the semester, we will have a good feeling for what the
answers are. Some of the characters that will appear (most of them somewhat
superficially) are:

·Projective
spaces.

·Elliptic
(and hyperelliptic) curves.

·Grassmannians.

·Jacobians.

·Rational
pointed curves.

·Riemann
Surfaces.

·Toric
Varieties.

·Quiver
Representations.

·Hilbert
Schemes.

While all moduli spaces are somewhat
geometric (they are spaces after all!), they are not at all exclusive domain of
algebraic geometry! They naturally feature in many other disciplines. Looking
at the list above you might recognize some of your favorite analytic,
arithmetic or combinatorial objects. I will try to keep a fairly ecumenical
point of view, that seems to me to be the most likely to be beneficial to a
somewhat diverse crowd as we’ll be.

Ideally the beginning of the
course will be soft (I would like to assume very little previous knowledge),
and the ending pretty steep (I would like to discuss some current research
topics). Hopefully you will help me with adjusting the slope in between.

NOTES:

I will give a
good effort at trying to keep my notes updated here. You are welcome to use
them, and you are even more welcome to contribute to them, by adding/
editing/integrating/modifying them etc. Just let me know if you want to mess
with them and we’ll set up a way to make that easy and natural.